She attended the fund-raiser 'at the invitation of' the committee chairperson. [=she attended because she was invited by the committee chairperson]
You have a 'standing/open invitation' [=you are always welcome; you are always invited] to visit us any time.
The event is 'by invitation only'. [=only people who have been invited to the event can attend it]
(..) except on special occasions. Not everyone interested in empathicalism meets him, any more than every American meets the President. An invitation to Flostre's home is a great honour, and as hard to get as an invitation to... The White House. I don't think jokes about Flostre (..)1957 Funny Face